1. Field of the Disclosure
This invention relates to apparatus which determines the performance characteristics of a pumping well. More particularly, the invention is directed to apparatus for determining downhole conditions of a sucker rod pump in a vertical borehole or deviated borehole from data which are received, measured and manipulated at the surface of the well. The invention also concerns the analysis of pumping problems in the operation of sucker rod pump systems in such boreholes. A vertical borehole is one that is substantially vertical into the earth, but a deviated borehole is one that is non-vertical into the earth from the surface. A deviated borehole may be a horizontal borehole which extends from a vertical portion thereof.
Still more particularly, the invention concerns an improved controller for analysis of downhole pump performance of a deviated borehole over the methods described in prior methods developed for nominally vertical borehole as described in Gibbs' U.S. Pat. No. 3,343,405 of Sep. 26, 1967.
2. Description of Prior Art
For pumping deep wells, such as oil wells, a common practice is to employ a series of interconnected rods for coupling an actuating device at the surface with a pump at the bottom of the well. This series of rods, generally referred to as the rod string or sucker rod, has the uppermost rod extending up through the well casinghead for connection with an actuating device, such as a pump jack of the walking beam type, through a coupling device generally referred to as the rod hanger. The well casinghead includes means for permitting sliding action of the uppermost rod which is generally referred to as the “polished rod.”
FIG. 1 depicts a prior art rod pumping well, illustrated for a nominally vertical borehole. FIG. 2 depicts a prior art surface measurement arrangement by which a surface dynamometer (“card”) is measured.
FIG. 1, shows a nominally vertical well having the usual well casing 10 extending from the surface to the bottom thereof. Positioned within the well casing 10 is a production tubing 11 having a pump 12 located at the lower end. The pump barrel 13 contains a standing valve 14 and a plunger or piston 15 which in turn contains a traveling valve 16. The plunger 15 is actuated by a jointed sucker rod 17 that extends from the piston 15 up through the production tubing to the surface and is connected at its upper end by a coupling 18 to a polished rod 19 which extends through a packing joint 20 in the wellhead.
FIG. 2, shows that the upper end of the polished rod 19 is connected to a hanger bar 23 suspended from a pumping beam 24 by two wire cables 25. The hanger bar 23 has a U-shaped slot 26 for receiving the polished rod 19. A latching gate 27 prevents the polished rod from moving out of the slot 26. A U-shaped platform 28 is held in place on top of the hanger bar 23 by means of a clamp 29. A similar clamp 30 is located below the hanger bar 23. A strain-gauge load cell 33 is bonded to the platform 28. An electrical cable 34 leads from the load cell 33 to an on-site well manager 50. A taut wire line 36 leads from the hanger bar 23 to a displacement transducer 37 (See FIG. 1). The displacement transducer 37 is also connected to the well manager 50 by the electrical lead 36′.
The strain-gauge load cell 33 is a conventional device and operates in a manner well known to those in the art. When the platform 28 is loaded, it becomes shorter and fatter due to a combination of axial and transverse strain. Since the wire of a strain-gauge 28 is bonded to the platform 28, it is also strained in a similar fashion. As a result, a current passed through the strain-gauge wire now has a larger cross section of wire in which to flow, and the wire is said to have less resistance. As the hanger bar 23 moves up and down, an electrical signal which relates strain-gauge resistance to polished rod load is transmitted from the load cell 33 to the well manager 50 via the electrical cable 34.
The displacement transducer 37 is a conventional device and operates in a manner well known to those of skill in the art of instrumentation. The displacement transducer unit 37 is a cable-and-reel driven, infinite resolution potentiometer that is equipped with a constant tension (“negator” spring driven) rewind assembly. As the hanger bar 23 moves up and down, the taut wire line 36 actuates the reel driven potentiometer and a varying voltage signal is produced. This signal, relates voltage to polished rod displacement, is also transmitted to the well manager 50. Other means for obtaining a displacement signal are well known in the art of determining performance characteristics of a pumping well.
Well manager 50 records the displacement signal as a function of time along with the rod load signal as a function of time.
In deep wells the long sucker rod has considerable stretch, distributed mass, etc., and motion at the pump end may be radically different from that imparted at the upper end. In the early years of rod pumping production, the polished rod dynamometer provided the principal means for analyzing the performance of rod pumped wells. A dynamometer is an instrument which records a curve, usually called a “card,” of polished rod load versus displacement. The shape of the curve or “card” reflects the conditions which prevail downhole in the well. Hopefully the downhole conditions can be deduced by visual inspection of the polished rod card or “surface card.” Owing to the diversity of card shapes, however, it was frequently impossible to make a diagnosis of downhole pump conditions solely on the basis of visual interpretation. In addition to being highly dependent on the skill of the dynamometer analyst, the method of visual interpretation only provides downhole data which are qualitative in nature. As a result it was frequently necessary to use complicated apparatus and procedures to directly take downhole measurements in order to accurately determine the performance characteristics at various depth levels within the well.
In 1936 W. E. Gilbert and S. B. Sargent disclosed an instrument which literally directly measured a subsurface dynamometer card. It was a mechanical device which was first run above the pump in the rod string. It allowed a small number of dynamometer cards to be collected before being recovered by pulling the rods to the surface. It scribed the pump card on a rotating tube, the angular position of which was made proportional to plunger position with respect to the tubing. Pump load was measured as proportional to the stretch of a calibrated rod within the instrument. Because the sucker rod had to be pulled to record the pump cards, the instrument was costly and cumbersome to use. But it provided valuable information relating the shape of the pump cards to various operating conditions known to exist in pumping wells such as full fillage, gas interference, fluid pound, pump malfunction, etc. The quantitative data that it provided allowed improvement of the methods for predicting pump stroke and the volumetric capability of the pump. The pump dynamometer device was a development that paved the way in the history of rod pumping technology.
With the dawn of the digital computer, S. G. Gibbs, a co-inventor of this invention, patented in 1967 (U.S. Pat. No. 3,343,409) a method for determining the downhole performance of a rod pumped well by measuring surface data, (the surface card) and computing a load versus displacement curve (a “pump card” for the sucker rod string at any selected depth in the well). As a result, the system provided a rational, economical, quantitative method for determining downhole conditions which is independent of the skill and experience of the analyst. It was no longer necessary to guess at downhole operating conditions on the basis of recordings taken several thousands of feet above the downhole pump at the polished rod at the surface, or to undertake the expensive and time consuming operation of running an instrument to the bottom of the well in order to measure such conditions. By use of the method, it became possible to directly determine the subsurface conditions from data received at the top of the well.
The 1967 patent, U.S. Pat. No. 3,343,409 of Gibbs, showed that an analysis of rod pumping performance begins with an accurate calculation of the downhole pump card. Gibbs showed that the calculation is based on a boundary-value problem comprising a partial differential equation and a set of boundary conditions.
The sucker rod is analogous mathematically to an electrical transmission or communication line, the behavior of which is described by the viscously damped wave equation:
                                                        ∂              2                        ⁢                          u              ⁡                              (                                  x                  ,                  t                                )                                                          ∂                          t              2                                      =                                            v              2                        ⁢                                                            ∂                  2                                ⁢                                  u                  ⁡                                      (                                          x                      ,                      t                                        )                                                                              ∂                                  x                  2                                                              -                      c            ⁢                                          ∂                                  u                  ⁡                                      (                                          x                      ,                      t                                        )                                                                              ∂                t                                              +          g                                    (        1        )            where:v=velocity of sound in steel in feet/second;c=damping coefficient, 1/second;t=time in seconds;x=distance of a point on the unrestrained rod measured from the polished rod in feet; and,u(x,t)=displacement from the equilibrium position of the sucker rod in feet,g=weight of pump rod assembly.
In reality, damping in a sucker rod system is a complicated mixture of many effects. The viscous damping law postulated in Equation 1 lumps all of these damping effects into an equivalent viscous damping term. The criterion of equivalence is that the equivalent force removes from the system as much energy per cycle as that removed by the real damping forces.
FIG. 1 shows that a pump 200 can be controlled based on a downhole “pump” card. U.S. Pat. No. 5,252,031 to S. G. Gibbs illustrates generation of control signals based on pump card determination. U.S. Pat. No. 6,857,474 by Bramlett et al. describes control of a pump based on pattern recognition of a pump card to analyze pump operation and control thereof. Such patents are incorporated by reference herein.
The wave equation, a second order partial differential equation in two independent variables (distance x and time t), models the elastic behavior of a long, slender rod such as used in rod pumping. As discussed in SPE paper 108762 titled, “Modeling a Finite Length Sucker Rod Using the Semi-Infinite Wave Equation and as Proof to Gibbs' Conjecture,” SPE 2007 Annual Technical Conference, Anaheim, Calif., 11-14, November 2007, J. J. DaCunha and S. G. Gibbs. Normally the problem to be solved with the wave equation involves boundary conditions specifying position at the top, and strain at the top and bottom of the rod string,
                                          u            ⁡                          (                              0                ,                t                            )                                =                      P            ⁡                          (              t              )                                      ,                                            α              ⁢                                                          ⁢                              u                ⁡                                  (                                      L                    ,                    t                                    )                                                      +                          β              ⁢                                                ∂                  u                                                  ∂                  x                                            ⁢                              (                                  L                  ,                  t                                )                                              =                      J            ⁡                          (              t              )                                      ,        α        ,                  β          ∈          R                ,                            (        2        )            
together with two conditions specifying initial position and velocity,
                                          u            ⁡                          (                              x                ,                0                            )                                =                      f            ⁡                          (              x              )                                      ,                                                            ∂                u                                            ∂                t                                      ⁢                          (                              x                ,                0                            )                                =                      g            ⁡                          (              x              )                                                          (        3        )            along the rods. For the sucker rod problem the damping law in the wave equation was chosen primarily for mathematical tractability even though it did not rigorously mimic the real dissipation effects along the sucker rod.
The boundary value problem that led to computation of downhole pump cards is incompletely stated. The initial conditions in Equation (3) above are ignored. It is presumed that friction damps out the initial transients, and the steady state behavior of the rod string is the same regardless of how the pumping system is started. No assumptions are made about conditions at the downhole pump. After all, determination of these conditions is the object of the solution. Thus, no boundary conditions analogous to Equation (2) above are specified at the pump. Instead, two boundary conditions are enforced at the surface,
                                          u            ⁡                          (                              0                ,                t                            )                                =                      P            ⁡                          (              t              )                                      ,                              EA            ⁢                                          ∂                u                                            ∂                x                                      ⁢                          (                              L                ,                t                            )                                =                      L            ⁡                          (              t              )                                      ,                            (        4        )            where E and A are the Young's modulus and the cross-sectional area of the rod string, respectively. Using digital methods, the time histories P(t) and L(t) are sampled at equal time increments and expressed as truncated Fourier series
                                          P            ⁡                          (              t              )                                =                                    φ              0                        +                                          ∑                                  n                  =                  1                                m                            ⁢                                                φ                  n                                ⁢                                  cos                  ⁡                                      (                                          n                      ⁢                                                                                          ⁢                      ω                      ⁢                                                                                          ⁢                      t                                        )                                                                        +                                          δ                n                            ⁢                              sin                ⁡                                  (                                      n                    ⁢                                                                                  ⁢                    ω                    ⁢                                                                                  ⁢                    t                                    )                                                                    ,                            (        5        )            
                              L          ⁡                      (            t            )                          =                              σ            0                    +                                    ∑                              n                =                1                            m                        ⁢                                          σ                n                            ⁢                              cos                ⁡                                  (                                      n                    ⁢                                                                                  ⁢                    ω                    ⁢                                                                                  ⁢                    t                                    )                                                              +                                    τ              n                        ⁢                                          sin                ⁡                                  (                                      n                    ⁢                                                                                  ⁢                    ω                    ⁢                                                                                  ⁢                    t                                    )                                            .                                                          (        6        )            
Using separation of variables, solutions to the wave equation are sought which satisfy the measured time histories of surface position and load. The resulting solutions for rod position and rod load, i.e.
                                          u            ⁡                          (                              x                ,                t                            )                                ⁢                                          ⁢          and          ⁢                                          ⁢          EA          ⁢                                    ∂              u                                      ∂              x                                ⁢                      (                          x              ,              t                        )                          ,                            (        7        )            
respectively, are evaluated at a specific depth and at a succession of times to produce the downhole pump card. See for example the computed card in a 5175 ft well shown in FIG. 3. The illustration also shows the measured surface data (in conventional dynamometer card form) from which the pump card is deduced. The method of computing downhole pump cards with the wave equation is described in the Gibbs patent referenced above. FIG. 3 shows prior art surface and pump card plots for a vertical well using the Gibbs method of calculating the pump card based on the surface card measured data.
Using empirical evidence, the wave equation solution outlined above was conjectured to be valid in spite of theoretical questions surrounding the incompletely stated problem from whence it came. It could be used to determine conditions at the pump if the friction law incorporated into the wave equation was correct. The conjecture is formally stated as the Gibbs' Conjecture.                Solutions of the wave equation which match measured time histories of surface load and position will produce the exact downhole pump card if the friction law in the wave equation is perfect. In computing the pump card, no knowledge of pump conditions is required. Any error in the friction law will cause error in the computed pump card.        
The paper (SPE 108762) mentioned above shows a non-constructive mathematical proof that downhole conditions in a finite rod string can be inferred from measurements at the top of a semi-infinite rod. The proof is developed by realizing that the laws of physics demand that information about down-hole pump conditions propagate to the surface in the form of stress waves. A key element in the proof, (and now the Gibbs' Theorem) is that the exact law of rod friction must be known. Even though the non-constructive proof does not reveal the exact law, the proof does show how the process can be used to refine the friction law to attain more accuracy in computing downhole conditions.
The term
  c  ⁢            ∂              u        ⁡                  (                      x            ,            t                    )                            ∂      t      is the fluid friction term representing the opposing force of the fluid against axial motion of the pump. In its simplest form, it prescribes a frictional force that is proportional to speed. No other rod frictional forces are presumed to exist. The g term represents rod weight. In other words the mathematical modeling of a rod pump as described by equation (1) presumes a nominally vertical well where tubing drag forces are assumed not to exist.
The qualifying word nominally is used because it is impossible to drill a perfectly vertical well. As weight is applied on the bit to achieve penetration, the drill string buckles somewhat and the borehole departs somewhat from the vertical. When a well is intended to be vertical, the oil producer includes a deviation clause in the agreement with the drilling contractor stipulating that the borehole be vertical within narrow limits. Vertical wells are easier to produce with rod pumping equipment because rod friction is less. The rod string transmits energy from the surface unit to the down hole pump which lifts fluid to the surface. Friction causes a loss in pump stroke and as a result decreases lifting capacity. Also it causes wear and tear on rods and tubing.
The practice of including deviation clauses in drilling contracts and the technology of measuring borehole path came about because of scandals in the oil industry. Unscrupulous oil producers were intentionally draining oil reserves owned by neighboring leaseholders using slanted wells.
Deviated wells are becoming more common. In these wells, the point where (in plan view) fluid from the reservoir enters the borehole can be considerably displaced laterally from the surface location. The deviation can be unintended or intentional as described above.
The reasons for intentionally deviated wells are many and varied. Most reasons follow from environmental or social considerations. Along a shoreline, wells with onshore surface locations can be deviated to drain reservoirs beneath bodies of water. Similarly oil beneath residential or metropolitan areas can be produced with deviated wells having their surface locations outside the sensitive areas. Oil and gas production requires vehicular traffic to service the wells. Deviated wells can diminish unwanted traffic in residential areas because only the surface locations need be serviced. The reach of deviated wells can be thousands of feet (in plan view) from the surface location. Multiple vertical wells require multiple surface roads to each location. A case in point could be ANWAR (Arctic National Wildlife Refuge). Using deviated wells, access roads to each well would not be necessary. Twenty or more deviated wells can be clumped together in a small area so as to produce a minimal environmental impact. A single access road to the small surface location would then suffice. Twenty different access rods to each well (if drilled vertically) would not be needed. Owing to these many reasons, the number of deviated wells has (and will continue to) increase rapidly.
Measuring and controlling the borehole path has become very sophisticated. Various telemetry methods are used to transmit triplets of data (depth, azimuth and inclination) to the surface. These are the items required to produce a deviation survey.